It's happening both ways. The finance and finance-related things are all downtown and the inner-loop is gentrifying. So is the East End (in a BIG way).
BUT because the oil companies are moving their operations west (BP and others) and north (Exxon) people are also migrating those directions.
You would think this would mean that real-estate prices in the donut between the loop and the beltway would be pushed downward but it's not happening. The population seems to be growing faster than the "city" expands so prices are going up everywhere.